Application of Gauss - Lanczos Algorithm to Determine Low Modes Density of Dirac Operator

  • Rudina Osmanaj Department of Physics, Faculty of Natural Sciences, University of Tirana
  • Dafina Xhako Department of Physics Engineering, Faculty of Mathematical Engineering and Physical Engineering, Polytechnic University of Tirana
  • Niko Hyka Department of Diagnostics, Faculty of Medical Technical Sciences, Medical University of Tirana, Tirana
Keywords: Boriçi - Creutz operator, eigenvalues, Dirac operator, Gauss-Lanczos quadrature, lattice QCD, numerical simulation

Abstract

There are numerous applications in physics, especially in Lattice QCD, where is required to bound entries and the trace of the inverse and the determinant of a large sparse matrix. This paper review one of the most popular methods which are used in lattice QCD to compute the determinant of the lattice Dirac operator: Gaussian integral representation. A modified algorithm can be used for other purposes too, for example for the determination of the density of eigenvalues of the Dirac operator near the origin. This because in Lattice QCD, low-lying Dirac modes are a suitable tool to understand chiral symmetry since they encode the nature of quark propagation as well as the condensate itself in the chiral regime. The formation of a non-zero chiral condensate is an effect of the accumulation of the low modes of the Dirac operator near zero. We review the development in Krylov subspace evaluation of matrix functions and we develop a practical numerical algorithm to achieve a reliable determination of the density of eigenvalues of the Dirac operator near the origin using the Gauss-Lanczos quadrature. We utilize the optimal properties of Krylov subspaces in approximating the distribution of the eigenvalues of the Dirac operator. In this work we have used the Boriçi - Creutz operator to test our method, as an example of using this algorithm in Lattice QCD.

References

[1] Fahey G Bai Zh and Golub G. Some large-scale matrix computation problem. In: Journal of Computational and Applied Mathematics. vol. 741. no. 2. 1996. pp. 71–89.
[2] Banks T and Casher A. Chiral symmetry breaking in confining theories. In: Nuclear Physics B. vol. 169. no.1. 1980. pp. 103–125.
[3] Bedaque et al. P. Broken symmetries from minimally doubled fermions. In: Physics Letters B, vol. 662, no. 5, 2008, pp. 449–455.
[4] Boriçi A. Minimally Doubled Fermion Revival. In: PoS LATTICE2008, 2008, p. 231.
[5] Capitani et al. S. Renormalization of minimally doubled fermions. In: Journal of High Energy Physics, vol.9, 2010, pp. 27-32.
[6] Creutz M. Four-dimensional graphene and chiral fermions. In: Journal of High Energy Physics, vol..4, 2008, pp. 017-022.
[7] Damgaard et al. P H. The microscopic spectral density of the QCD Dirac operator. In: Nuclear Physics B, vol. 547, no.1, 1999, pp 305–328.
[8] Fukaya et al. H. Determination of the chiral condensate from QCD Dirac spectrum on the lattice. In: Physical review D, vol. 83, no. 7, 2011, pp. 074501.
[9] Giusti L and Lüscher M. Chiral symmetry breaking and the Banks-Casher relation in lattice QCD with Wilson quarks. In: Journal of High Energy Physics, vol.03, 2009, p. 013.
[10] Golub G and Van Loan C. Matrix computations. In: JHU Press. 2013. pp. 105-153.
[11] Karsten L. Lattice fermions in Euclidean space-time. In: Physics Letters B. vol. 104. no.4. 1981. pp. 315–319.
[12] Kogut J and Susskind L. Hamiltonian formulation of Wilson’s lattice gauge theories. In: Physical Review D. vol. 11. no. 2. 1975. pp. 395-342.
[13] Lanczos C. Solution of systems of linear equations by minimized iterations. In: J. Res. Nat. Bur. Standards. vol. 49. no.1. 1952. pp. 33–53.
[14] Nambu Y and Jona-Lasinio G. Dynamical model of elementary particles based on an analogy with superconductivity. In: Physical Review. vol. 122. no. 1. 1961. pp. 345-354.
[15] Neuberger H. Overlap lattice Dirac operator and dynamical fermions. In: Physical Re- view D. vol. 60. no.6. 1999. p. 065006.
[16] Nielsen H. and Ninomiya M. Absence of neutrinos on a lattice: (I). Proof by homotopy theory. In: Nuclear Physics B. vol. 185. no. 1. 1981. pp. 20–40.
[17] Osborn J. Toublan D. Verbaarschot M. From chiral random matrix theory to chiral perturbation theory. In: Nuclear Physics B. vol. 540. no.1. 1999. pp. 317–344.
[18] Smilga A and Stern J. On the spectral density of Euclidean Dirac operator in QCD. In: Physics Letters B. vol. 318. no.3. 1993. pp. 531–536.
[19] Wilczek F. Lattice fermions. In: Physical review letters, vol. 59. no.21. p. 2397. 1987.
[20] Wilson K. Confinement of quarks. In: Physical Review D. vol. 10. no.8. 1974. p. 2445.
[21] Capitan S. Creutz M. Weber J. Wittig H. Numerical studies of Minimally Doubled Fermions. In: Journal of High Energy Physics. vol. 10. no.9. 2010. p.27.
[22] Di Pierro M. Matrix Distributed Processing and FermiQCD. In: Journal of High Energy Physics. 2000. hep-lat/0011083.
[23] Parman S. and Machmudah A. Waveform based Inverse Kinematics Algorithm of Kinematically Redundant 3- DOF In: International Journal of Innovative Technology and Interdisciplinary Sciences. ISSN: 2613 – 7305. DOI: https:// doi.org/10.15157/IJITIS.2020.3.2.407-428. Volume 3. Issue 2. 2020. pp. 407- 428.
[24] Hyka-Xhako D. Osmanaj R. Jani J. The physical scale from string tension in LQCD The 10th Jubilee Conference of the Balkan Physical Union. BPU10 – August 2018. Bulgary. Proceedings in AIP Conference Proceedings 2075. https://doi.org/10.1063/1.5091253. eISSN: 1551-7616. 2019. 110002.
[25] Xhako D and Zeqirllari R. Chiral fermions algorithms in lattice QCD. In: East European Journal of Physics (East. Eur. J. Phys). DOI:10.26565/2312-4334-2019-1-02. Volume 1. 2019. 34-39.
[26] Zeqirllari R. and Xhako D. Minimally doubled fermions and spontaneous chiral symmetry breaking. In: EPJ (European Physics Journal) Web of Conferences 175. https://doi.org/10.1051/epjconf/201817504002. 2018 pp.04002.
[27] Zeqirllari R. and Xhako D. Preliminary results of the hypercubic symmetry restauration of Boriçi – Creutz fermions in the light hadrons spectrum. In: AIP (American Institute of Physics) Conference Proceedings 2075. https://doi.org/10.1063/1.5091257. 2019. pp. 110006.
Published
2020-05-28
How to Cite
Osmanaj, R., Xhako, D., & Hyka, N. (2020). Application of Gauss - Lanczos Algorithm to Determine Low Modes Density of Dirac Operator. International Journal of Innovative Technology and Interdisciplinary Sciences, 3(2), 443-450. https://doi.org/10.15157/IJITIS.2020.3.2.443-450